The second lower Loewy term of the principal indecomposable of a modular group algebra (Q5937405)

Let \(K=\mathbb{F}_p\) be the prime field of characteristic \(p\). Furthermore, let \(J\) be the Jacobson radical of the group algebra \(KG\) and \(P\) the projective cover of the trivial module. If \(G\) is \(p\)-solvable then Gaschütz completely described the second Loewy layer of \(P\), i.e. \(PJ/PJ^2\). More precisely, he proved that the composition factors of \(PJ/PJ^2\) are the complemented \(p\)-chief factors \(V\) of \(G\) counted with multiplicities, say \(\text{cm}(V)\). More generally, these \(p\)-chief factors occur with multiplicities as composition factors of \(PJ/PJ^2\) for any finite group \(G\), and if they are the only ones then \(G\) has to be \(p\)-solvable; a result of the reviewer [Commun. Algebra 13, 2433-2447 (1985; Zbl 0575.20012)]. The paper under review deals with a description of \(PJ/PJ^2\) for an arbitrary group \(G\). If \(V\) is a simple \(KG\)-module and \(l^G_2(V)\) denotes the multiplicity of \(V\) as a composition factor in \(PJ/PJ^2\) then NEWLINE\[NEWLINEl^G_2(V)=\text{cm}(V)+l^{G/C_G(V)}_2(V).NEWLINE\]NEWLINE So as a main problem it remains to determine \(l^{G/C_G(V)}_2\). The authors give an explicit description of \(PJ/PJ^2\) which is rather complicated and depends on simple sections of \(G\) which arise in a reduction process developed by Kovács.NEWLINENEWLINENEWLINEFor a general field \(K\) of prime characteristic \(p\) the answer is easily obtained by considering Galois conjugates.